Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
On condition numbers and the distance to the nearest III-posted problem
Numerische Mathematik
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Polynomial roots from companion matrix eigenvalues
Mathematics of Computation
Matrix computations (3rd ed.)
Applied numerical linear algebra
Applied numerical linear algebra
Error Bounds for Zeros of a Polynomial Based Upon Gerschgorin's Theorems
Journal of the ACM (JACM)
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Polynomial root finding using iterated Eigenvalue computation
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Null space and eigenspace computations with additive preprocessing
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Structured matrix methods for polynomial root-finding
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Additive preconditioning and aggregation in matrix computations
Computers & Mathematics with Applications
Computing clustered close-roots of univariate polynomials
Proceedings of the 2009 conference on Symbolic numeric computation
Real and complex polynomial root-finding with eigen-solving and preprocessing
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Root-finding by expansion with independent constraints
Computers & Mathematics with Applications
Extended companion matrix for approximate GCD
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Polynomial root-finding via structured matrices
ACM Communications in Computer Algebra
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We discuss an iterative algorithm that approximates all roots of a univariate polynomial. The iteration is based on floating-point computation of the eigenvalues of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about logρ/εχ(P) iterations, where ∈ is the relative error of floating-point arithmetic, ρ is the relative separation of the roots, and χ(P) is the condition number of the polynomial. Each iteration requires an n × n floating-point eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floating-point accuracy at up to n points.We describe a careful implementation of the algorithm, including many techniques that contribute to the practical efficiency of the algorithm. On some hard examples of ill-conditioned polynomials, e.g. high-degree Wilkinson polynomials, the implementation is an order of magnitude faster than the Bini-Fiorentino implementation mpsolve.